THE GREAT ALGOBIST
told "him in how many ways any given integer may be expressed
as such a sum.*
For those whose tastes are more practical we may cite
Jacobi's work in dynamics. In this subject, of fundamental
importance in both applied science and mathematical physics,
Jacobi made the first significant advance beyond Lagrange and
Hamilton. Readers acquainted with quantum mechanics will
recall the important part played hi some presentations of that
revolutionary theory by the Hamilton-Jacobi equation. His
work in differential equations began a new era.
In algebra, to mention only one thing of many, Jacobi cast
the theory of determinants into the simple form now familiar
to every student in a second course of school algebra.
To the Newton-Laplace-Lagrange theory of attraction
Jacobi made substantial contributions by his beautiful investi-
gations on the functions which recur repeatedly in that theory
and by applications of elliptic and Abelian functions to the
attraction of ellipsoids.
Of a far higher order of originality is his great discovery in
Abelian functions. Such functions arise in the inversion of an
Abelian integral, in the same way that the elliptic functions
arise from the inversion of an elliptic integral. (The technical
terms were noted earlier hi this chapter.) Here he had nothing
to guide him, and for long he wandered lost in a maze that had
no clue. The appropriate inverse functions in the simplest case
are functions of two variables having Jour periods; in the
general case the functions have n variables and 2n periods; the
elliptic functions correspond to n = 1. This discovery was to
nineteenth-century analysis what Columbus' discovery of
America was to fifteenth-century geography.
Jacobi did not suffer an early death from overwork, as his
lazier friends predicted that he would, but from smallpox
(18 February 1851) in his forty-seventh year. In taking leave
of this large-minded man we may quote his retort to the great
French mathematical physicist Fourier, who had reproached
* If n is odd, the number of ways is 8 times the sum of all the
divisors of n (1 and n included); if n is even, the number of ways is 24*
times the sum of all the odd divisors of n.